7 Non-asymptotic-time Dynamics
127
7.6 Conclusion
We have studied slowly time-varying one-dimensional phase dynamics to illustrate
the physical limitations of traditional mathematical formalisms of dynamical systems
analysis that have their roots in the study of isolated systems. For processes subject to
slowly time-dependent forcing, we have proposed an alternative framework of “slowfast finite-time dynamical systems” for mathematical analysis of physical concepts
such as stability. More generally, we argue that a finite-time perspective on dynamical
behaviour may be crucial to the correct identification and explanation of dynamical
phenomena occurring in open systems studied throughout the sciences.
Acknowledgements We are grateful for the comments of three anonymous referees, which have
led to considerable improvement of this manuscript. This study has been supported by an EPSRC
Doctoral Prize Fellowship, the DFG grant CRC 701, the EPSRC grant EP/M006298/1, and the
European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement No 642563. Codes for the numerics carried out in this chapter are available
upon request and are deposited on the Lancaster Publications and Research system Pure.
References
1. L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics (Springer,
Berlin, 1995)
2. A. Berger, T.S. Doan, S. Siegmund, Nonautonomous finite-time dynamics. Discret. Contin.
Dyn. Sys. B 9(3 & 4), 463–492 (2008)
3. L.H. Duc, J.P. Chávez, T.S. Doan, S. Siegmund, Finite-time Lyapunov exponents and metabolic
control coefficients for threshold detection of stimulus-response curves. J. Biol. Dyn. 10(1),
379–394 (2016)
4. R.M. Dudley, Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics.
Cambridge University Press (1999)
5. P. Giesl, J. McMichen, Determination of the area of exponential attraction in one-dimensional
finite-time systems using meshless collocation. Discret. Contin. Dyn. Sys. B 23(4), 1835–1850
(2018)
6. A. Glutsyuk, L. Rybnikov, On families of differential equations on two-torus with all phase-lock
areas. Nonlinearity 30(1), 61–72 (2016)
7. J. Guckenheimer, Y.S. Ilyashenko, The duck and the devil: canards on the staircase. Mosc.
Math. J. 1(1), 27–47 (2001)
8. Z. Hagos, T. Stankovski, J. Newman, T. Pereira, P.V.E. McClintock, A. Stefanovska, Synchronization transitions caused by time-varying coupling functions. Philos. T. R. Soc. A 377(2160),
20190275 (2019)
9. G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos
10(1), 99–108 (2000)
10. G. Haller, Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47(1), 137–162 (2015)
11. G. Haller, G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence.
Phys. D 147(3), 352–370 (2000)
12. Y.S. Ilyashenko, D.A. Ryzhov, D.A. Filimonov, Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations. Funct. Anal. Appl. 45(3), 192
(2011)
13. S. Ivi´ c, I. Mrša Haber, T. Legovi´ c, Lagrangian coherent structures in the Rijeka Bay current
field. Acta Adriat. 58(3), 373–389 (2017)
127
7.6 Conclusion
We have studied slowly time-varying one-dimensional phase dynamics to illustrate
the physical limitations of traditional mathematical formalisms of dynamical systems
analysis that have their roots in the study of isolated systems. For processes subject to
slowly time-dependent forcing, we have proposed an alternative framework of “slowfast finite-time dynamical systems” for mathematical analysis of physical concepts
such as stability. More generally, we argue that a finite-time perspective on dynamical
behaviour may be crucial to the correct identification and explanation of dynamical
phenomena occurring in open systems studied throughout the sciences.
Acknowledgements We are grateful for the comments of three anonymous referees, which have
led to considerable improvement of this manuscript. This study has been supported by an EPSRC
Doctoral Prize Fellowship, the DFG grant CRC 701, the EPSRC grant EP/M006298/1, and the
European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement No 642563. Codes for the numerics carried out in this chapter are available
upon request and are deposited on the Lancaster Publications and Research system Pure.
References
1. L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics (Springer,
Berlin, 1995)
2. A. Berger, T.S. Doan, S. Siegmund, Nonautonomous finite-time dynamics. Discret. Contin.
Dyn. Sys. B 9(3 & 4), 463–492 (2008)
3. L.H. Duc, J.P. Chávez, T.S. Doan, S. Siegmund, Finite-time Lyapunov exponents and metabolic
control coefficients for threshold detection of stimulus-response curves. J. Biol. Dyn. 10(1),
379–394 (2016)
4. R.M. Dudley, Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics.
Cambridge University Press (1999)
5. P. Giesl, J. McMichen, Determination of the area of exponential attraction in one-dimensional
finite-time systems using meshless collocation. Discret. Contin. Dyn. Sys. B 23(4), 1835–1850
(2018)
6. A. Glutsyuk, L. Rybnikov, On families of differential equations on two-torus with all phase-lock
areas. Nonlinearity 30(1), 61–72 (2016)
7. J. Guckenheimer, Y.S. Ilyashenko, The duck and the devil: canards on the staircase. Mosc.
Math. J. 1(1), 27–47 (2001)
8. Z. Hagos, T. Stankovski, J. Newman, T. Pereira, P.V.E. McClintock, A. Stefanovska, Synchronization transitions caused by time-varying coupling functions. Philos. T. R. Soc. A 377(2160),
20190275 (2019)
9. G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos
10(1), 99–108 (2000)
10. G. Haller, Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47(1), 137–162 (2015)
11. G. Haller, G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence.
Phys. D 147(3), 352–370 (2000)
12. Y.S. Ilyashenko, D.A. Ryzhov, D.A. Filimonov, Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations. Funct. Anal. Appl. 45(3), 192
(2011)
13. S. Ivi´ c, I. Mrša Haber, T. Legovi´ c, Lagrangian coherent structures in the Rijeka Bay current
field. Acta Adriat. 58(3), 373–389 (2017)